Hartree-Fock theory interaction energy

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

An observable energy for a selected set of electrons follows in a straightforward manner from the sum of their orbital energies less the electron-electron interactions, which are counted twice in this sum. This procedure, reflecting the spirit of Koopman s theorem, is nothing new in Hartree-Fock theory. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

Hence, the electronic energy is a functional ofthe density E i = rf[/>(r)] instead of a functional of the orbitals E = d[ ( / ( r )] (see Eq. (12.6) for the explicit expression in Hartree-Fock theory). However, the formally exact principles of DFT are connected with the drawback that the energy expression as a function of the density is not known. It can be approximated and is usually divided into different physically meaningful contributions (i.e. an electron-electron interaction energy functional is formulated, which is decomposed in a classical Poisson-type electrostatic interaction term and a rest term - the exchange-correlation functional -which is not known). 

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